User blog:Ecl1psed276/A list of all BMS versions and their differences
here you go Version 1 Created: ??? Creator: Bashicu Description: This version is the first version of BMS. It uses an extremely simple method of finding the bad root of an expression, a method that ultimately didn't work and caused an infinite loop. Terminates: No. An example of a non-terminating expression is (0,0)(1,1)(2,1)(3,1)(2,0)(1,1)(2,1)(3,1), and this one is believed to be the smallest one. Version 1.1 Created: ??? Creator: Bashicu Description: This version introduces the Upper branch ignoring model, which is the algorithm to find the bad root that is used in nearly all BMS versions after this one (except 2.2). All other rules are the same as BM1. Terminates: No. (0,0,0)(1,1,1)(2,0,0)(1,1,1) is an example of a non-terminating matrix, and it was found by Hyp cos. It is believed to be the smallest non-terminating matrix. Version 2 Created: ??? Creator: Bashicu Description: The second version of BMS made by Bashicu. It was created after BM1 was shown to have an infinite loop. The method used to find the bad root in BM2 is more complex than BM1, but it is believed to be the most logical and consistent way to do it. If BM2 terminates, this is one of the most powerful notations known, easily surpassing notations like SAN. Even (0,0,0)(1,1,1)(2,2,2) is believed to be beyond pretty much any notation in existence, except for TON. Terminates: No. For a long time, BM2 was believed to terminate, but after some studying, Nish and Koteitan have determined that BM2 is likely to have an infinite loop somewhere past (0,0,0,0)(1,1,1,1). TODO: Put an explicit counterexample here. Version 2.1 Created: ??? Creators: Koteitan Description: idk Terminates: No. Version 2.2 Created: ??? Creators: Koteitan Description: This version of BMS is the most unique of all the versions. It uses something called the "concestor method", a different method to find the bad root than BM2 uses. Analysis of this version looks identical to BM2 up to (0,0)(1,1)(2,0). However, analysis becomes extremely complicated from here on out, because sometimes adding a 1 to the ordinal doesn't actually correspond to adding a (0,0...0). Terminates: Unknown. Version 2.3 Created: July 18, 2018 Creators: Koteitan/Ecl1psed/Nish Description: After discovering an inconsistency in the expansion of a BM2 expression, this version was created by Nish and Koteitan. I also independently discovered this version when I tried to make a program for BM2. However, it turned out that my program didn't exactly correspond with BM2, it instead corresponded with the newly defined BM2.3. It works exactly like BM2 all the way up to (0,0,0,0)(1,1,1,1), but it does not have the same inconsistency that BM2 has. It is accepted by most of the community that BM2.3 is better and more logical than BM2. Terminates: Probably? Version 2.4 Created: June 24, 2018 Creators: Yukito Description: This version expands exactly like BM2.3, so we can treat the notations as equal for all intents and purposes. It's almost like Yukito independently discovered BM2.3, along with me and Nish/Koteitan. You just know that when 3 or 4 different people independently discover a notation at the same time that it is definitely a good notation. Terminates: Probably. Version 2.5 Created: July 20, 2018 Creators: Yukito Description: idk Terminates: idk Version 2.6 Created: around July 22, 2018 Creators: Nish Description: This version is based off of Nish's extension of primitive sequence system. It acts similarly to BM2 but is slightly more powerful. However, Nish thinks that BM2 catches up to this version at (0,0,0)(1,1,1)(2,2,1)(3,0,0). Terminates: Unknown. Version 3 Created: June 12, 2018 Creator: Bashicu Description: At the point of BM3's release, it was thought that BM2 was likely to terminate, so it was not known exactly why Bashicu decided to put this version out. I think the reason he did was to make a version of BM3 that has more consistent behaviour. Not entirely sure though. Terminates: No. Just mere days after it's inclusion into Fish's calculator, Nish/Alemagno12 found an infinite loop with the expression (0,0,0)(1,1,1)(2,1,1)(1,1,1). Version 3.1 Created: July 12, 2018 Creator: Nish Description: This version acts like BM2, except in the calculation of the C vectors, if a vector does not equal (1,1,1...1,1) with only 1's, then we set it to (1,0,0,0...0,0). (sorry this is not the place to explain what a C vector is). This modification makes the analysis MUCH easier than BM2. For example, (1,1,1) now always adds \(\Omega_\omega\) inside the psi function, whereas with BM2, adding a (1,1,1) sometimes adds \(\Omega_\omega\), but other times it changes a \(\Omega\) into \(\Omega_\omega\). However, this also weakens the notation slightly. It is not known if the notation ever catches up to BM2. Some people, like myself, think that it catches up at (0,0,0)(1,1,1)(2,2,2), and others think it will never catch up at all. Terminates: No. Version 3.1.1 Created: July 20, 2018 Creator: Ecl1psed (me) Description: why does this version even exist lol. In discord, I said my idea of a new version of BMS, based off BM3.1. However, within 5 minutes of sending that message, I realized that my idea would not work, because it would create expressions like (2,0)(4,0) which is obviously not allowed. But for some reason, everyone thought it would be a good idea to give my notation it's own version name, so here we are. Don't even get me started. Terminates: It's completely broken, so you can't even say whether it terminates or not lol. Version 3.2 Created: July 23, 2018 Creator: Nish Description: Remember how we got from BM2 to BM3.1? What if you instead start with BM2.3 instead of BM2? Then, you get this version, BM3.2. It is highly reminiscent of BM3.1. Terminates: Maybe? Version 3.3 Created: March 21, 2019 Creators: Ecl1psed, Rpakr Description: This version is an attempt to imitate IBMS (Idealized BMS). Regular BM2.3 has an issue with analysis where appending a (1,1,1) seems to have inconsistent effects. It's often equivalent to adding \(\Omega_\omega\) inside \(\psi(\alpha)\), but other times it has more powerful effects. IBMS is a hypothetical version of BMS where appending (1,1,1) always has the effect of adding \(\Omega_\omega\) inside \(\psi(\alpha)\). So far, no matrix is known whose expansion deviates from what it should be, according to IBMS. Terminates: Probably? If BM2.3 terminates, then BM3.3 almost certainly terminates as well. Version 4 Created: ??? Creators: Bashicu Description: This version is identical to BM2.3. When we need to talk about this notation, we will say BM2.3. Terminates: Probably? There are quite a lot of versions of Bashicu Matrix System. Out of these, the ones that are most likely to terminate are BM2.3 and BM3.3. The analysis of BM3.3 is much easier than BM2.3, but BM2.3 is more powerful. It is not known whether BM3.3 catches up to BM2.3 at some point.. Category:Blog posts